TR/06/87 April 1987

TR/06/87 April 1987

C

A Polygonal Surface Patch

by

John A. Gregory and Jörg M. Hahn

z1624820

C

A Polygonal Surface Patch

By

John A. Gregory and Jörg M. Hahn Department of Mathematics and Statistics

Brunel University Uxbridge UB8 3PH, England

T h i s w o r k w a s s u p p o r t e d b y t h e S c i e n c e a n d E n g i n e e r i n g R e s e a r c h C o u n c i l g r a n t GR/D/77148.

Presented at the meeting "Surfaces in Computer Aided Geometric Design", Mathematisches Forschungsinstitut Oberwolfach, February, 1987.

S u b m i t t e d f o r p u b l i c a t i o n i n C o m p u t e r A i d e d G e o m e t r i c D e s i g n .

A C

Polygonal Surface Patch

by John A. Gregory and Jörg M. Hahn

A b s t r a c t A p o l y g o n a l p a t c h i s d e f i n e d t o f i l l a n n - s i d e d h o l e w i t h i n a C

p a r a m e t r i c c o n t i n u o u s rectangular patch complex.

Key words

Polygonal Patches

-1- 1. I n t r o d u c t i o n

I n t h i s n o t e a p o l y g o n a l s u r f a c e p a t c h i n t e r p o l a n t i s p r e s e n t e d w h i c h w i l l f i l l a n n - s i d e d h o l e w i t h i n a C

p a r a m e t r i c c o n t i n u o u s r e c t a n g u l a r p a t c h c o m p l e x . T h e p o l y g o n a l p a t c h i n t e r p o l a n t i s b a s e d o n a c o n v e x c o m - b i n a t i o n c o n s t r u c t i o n , w i t h a p p r o p r i a t e c h o i c e o f t h e b o u n d a r y d a t a t o e n s u r e g e o m e t r i c c o n t i n u i t y o f o r d e r 2 w i t h t h e a d j o i n i n g r e c t a n g u l a r p a t c h e s ( GC

).

A m e t h o d f o r s o l v i n g t h e g e n e r a l GC ^k p r o b l e m i s p r e s e n t e d i n [ G r e g o r y a n d H a h n ‘ 8 7 ] . T h i s m e t h o d i n v o l v e s t h e C ^k r e p a r a m e t e r i z a t i o n o f t h e s u r f a c e a r o u n d t h e h o l e s o t h a t i t i s d e f i n e d o n t h e e x t e r i o r o f a r e g u l a r p o l y g o n . T h e m e t h o d f o r t h e p a r t i c u l a r C

c a s e p r o p o s e d h e r e i s d i f f e r e n t , h o w e v e r , i n t h a t i t i n v o l v e s t h e d i r e c t u s e o f g e o m e t r i c G C

c o n t i n u i t y c o n d i t i o n s i n t h e d e v e l o p m e n t o f t h e p a t c h .

W e w i l l m a k e u s e o f g e o m e t r i c c o n t i n u i t y c o n d i t i o n s b e t w e e n p a t c h e s i n t h e t o t a l d e r i v a t i v e f o r m g i v e n i n [ G r e g o r y a n d H a h n ‘ 8 6 ] . T h e p r e s e n t p a p e r c a n b e c o n s i d e r e d a s a s e q u e l t o t h i s e a r l i e r p a p e r i n t h a t i t p r o v i d e s a

s o l u t i o n t o t h e p o l y g o n a l p a t c h p r o b l e m p o s e d t h e r e . W e b e g i n b y b r i e f l y r e - v i e w i n g t h e c o n d i t i o n s f o r g e o m e t r i c c o n t i n u i t y b e t w e e n p a t c h e s . F u r t h e r details can be found in the earlier paper or in [Hahn ‘87] of these p r o c e e d i n g s , w h e r e a f u l l d i s c u s s i o n o f g e o m e t r i c c o n t i n u i t y i s g i v e n t o g e t h e r w i t h a p p r o p r i a t e r e f e r e n c e s .

2 . C o n d i t i o n s f o r G e o m e t r i c C o n t i n u i t y b e t w e e n P a t c h e s

L e t p : Ω p → I R ³ a n d q : Ω q → I R 3 be two parametric surface patches defined on closed polygonal domains Ω p ⊂ I R ² and Ω q ⊂ I R 2 . (We are specifically concerned with the case of Ω _q a rectangle and Ω p an n-sided polygon.) Assume that p a n d q a r e r e g u l a r C

m a p s a n d l e t e

: [ 0 , 1 ] → I R

a n d e

: [ 0 , 1 ] → I R

b e r e g u l a r

parametric representations of boundary segments of Ω _p and Ω _q respectively.

Then the following proposition is based on Lemma 2.1 of [Gregory and Hahn ‘ 8 6 ] ,

w h e r e , f o r n o t a t i o n a l c o n v e n i e n c e , w e h a v e r e v e r s e d t h e r o l e s o f p a n d q

f r o m t h e e a r l i e r p a p e r .

-2-

Proposition 1. The patches p and q have a GC

join across their respective boundary segments e _p and e _q , if there exists a C

diffeomorphism φ : I R

→ I R

, defined in a neighbourhood of e

, which is such that:

( i ) ( Domain continuation) e _q = φ o e _p and interior points of Ω _q are mapped from exterior points of Ω _p .

( i i ) ( Patch continuity) Given any (non-zero) transversal vector field U(s) defined on e _p ( s ) , then

( ) s i ( q ) e ( s ) ( U i ( s )), i 0 , 1 , 2 , for all 0 s 1 . U i

) s ( p e i

p

p

⎜ ⎝ ⎛ ⎟ ⎠ ⎞ = ∂ φ = ≤ ≤

∂ o (2.1)

Here, ∂ p _x is the fi rst d er i v a ti ve l i n ear map , ∂ ² p _x is the second derivative summetric bilinear map at x ∈ I R ² and denotes the i-tuple (U(s),...,U(s)). U ⁱ ( s )

Expanding the right hand sides of (2.1) using the chain and product rules, and writing

)), s ( U ), s ( U ( )

s ( W )), s ( U ( )

s (

V = ∂ φ _e

_{( s )} = ∂ ² φ _e

_{( s )} (2.2)

leads to the following proposition, the sufficiency of which is proved in [Gregory and Hahn ‘86], (see also [Hahn ‘87] for a weaker form of the proposition).

Proposition 2. Let U : [ 0 , 1 ] → I R

be a C

(non-zero) vector field which is t r a n s v e r s a l t o e

in an inward direction. Then the patches p and q have a GC join iff. there exist (i) a (non-zero)

C

vector field V : [ 0 , 1 ] → I R

,

tra nsversal to e

in an outward direction, and (ii) a C

vector field ,

R I ] 1 , 0 [ :

W →

which are such that

)), s q ( e ( q )) s p ( e (

p = (2.3)

( ) ( ) s ( U s ) q e ( ) ( ) s ( V s , p e

q

q

= ∂

∂ ) (2.4)

)) s ( W ) ( s ( q e )) s ( V ), s ( V ) ( s ( q e )) 2 s ( U ), s ( U ) ( s ( p e 2

q q

p

= ∂ + ∂

∂ (2.5)

Our final proposition is concerned with geometric continuity between p a t c h e s

p a n d q , w h e n p i s d e f i n e d a s a c o n v e x c o m b i n a t i o n o f t w o p a t c h e s p

a n d p

.

I t w a s o b s e r v e d i n [ G r e g o r y a n d H a h n ‘ 8 6 ] t h a t G C

c o n t i n u i t y b e t w e e n p

a n d q ,

a n d b e t w e e n p

a n d q , i s n o t s u f f i c i e n t t o e n s u r e GC

c o n t i n u i t y b e t w e e n t h e

p a t c h e s p and q. The following proposition, which gives a sufficient

-3-

c o n d i t i o n f o r a G C

j o i n , w i l l b e u s e d i n t h e d e v e l o p m e n t o f t h e p o l y g o n a l p a t c h .

P r o p o s i t i o n 3 . L e t p i : Ω p → I R 3 , i = 1 , 2 , h a v e G C

j o i n s w i t h q : Ω

→ I R

i n t h e s e n s e o f P r o p o s i t i o n 2 . T h u s

)) s q ( e ( q ) s p ( e i (

p = (2.6)

)) s i ( V ) ( s ( p e )) s ( U ) ( s ( e p i

q

p

= ∂

∂ (2.7) ))

s i ( W ) ( s ( q e )) s i ( V ), s i ( V ) ( s ( q e )) 2 s ( U )), s ( U ) ( s ( e p i 2

q q

p

= ∂ + ∂

∂ (2.8)

f o r a p p r o p r i a t e l y d e f i n e d v e c t o r f i e l d s U ( s ) , V

( s ) , W

( s ) , i = 1 , 2 . L e t

3

p

I R

:

p Ω → b e d e f i n e d b y

P ( x ) = W

( x ) P

( x ) + W

( x ) P

( x ) (2.9) w h e r e w i : Ω p → I R , i = 1 , 2 a r e C

f u n c t i o n s s u c h t h a t

. 2 , 1 j , ) 0 s ( ) e w 2 w 1 i ( and ) 1

s ( ) e w 2 w 1

( +

= ∂ +

= = (2.10)

T h e n a s u f f i c i e n t c o n d i t i o n t h a t p a n d q h a v e a GC

j o i n i s t h a t V

( s ) V ≡

( s ) .

T h e p r o o f o f t h i s p r o p o s i t i o n i s s t r a i g h t f o r w a r d s i n c e i t i s r e a d i l y s h o w n t h a t

)), s q ( e ( q )) s p ( e (

p =

)), s ( V ) ( s ( q e )) s ( U ) ( s ( p e

q

p

= ∂

∂

)) s ( V ), s ( V ) ( s ( q e )) 2 s ( U ), s ( U ) ( s ( p e 2

q

p

= ∂

∂ )),

s 2 ( W ) s 2 ( w ) s 1 ( W ) s 1 ( w ) ( s (

q e

+

∂ +

w h e r e w i ( s ) = w i ( e q ( s )) a n d V ( s ) ≡ V 1 ( s ) ≡ V 2 ( s ) . T h e e s s e n t i a l p o i n t t o n o t e

i n P r o p o s i t i o n 3 i s t h a t w e h a v e a s s u m e d t h e p a r t i c u l a r c a s e w h e r e V 1 ( s ) ≡ V 2 ( s ) a n d h e n c e t h a t p

a n d q , i = 1 , 2 , h a v e i d e n t i c a l G C

j o i n s .

3 . T h e P o l y g o n a l P a t c h P r o b l e m

L e t q

, i = 0 , . . . , n - 1 , f o r m a C

p a r a m e t r i c c o n t i n u o u s r e c t a n g u l a r p a t c h

c o m p l e x a r o u n d a n n - s i d e d h o l e . M o r e s p e c i f i c a l l y , l e t q i : [ 0 , 2 ] × [ 0 , 1 ] → I R 3

b e s u c h t h a t

-4-

q i ( s , t ) t 1 , j 0 , 1 , 2 , t j

: j ) 1 , s i ( j q ,

0 = =

∂

= ∂

∂

(3.1)

d e f i n e s t h e b o u n d a r y d a t a a l o n g t h e i ’ t h s e g m e n t o f t h e h o l e f o r 0 ≤ s ≤ 1 , s e e F i g u r e 1 . ( I n p r a c t i c e , e a c h q

m a y c o n s i s t o f a s u b - c o m p l e x o f r e c t a n g u l a r p a t c h e s b u t f o r d e s c r i p t i v e p u r p o s e s i t i s c o n v e n i e n t t o r e p r e s e n t t h i s a s o n e p a r a m e t r i c s u r f a c e . )

[0,2] x [0,l]

Figure 1

-5-

W e a s s u m e t h a t t h e p a t c h e s q

, i = 0 , . . . , n - 1 , f o r m a C

p a r a m e t r i c a l l y continuous patch complex in the sense that the composed map

(3.2)

⎩ ⎨

⎧

×

∈

×

−

∈ +

− −

→ q i ( s , t ) , ( s , t ) [ 0 , 2 ] [ 0 , 1 ]

] 2 , 0 [ ] 0 , 1 [ ) t , s ( , ) s 1 , t 2 1 ( q i ) t , s (

i s C

c o n t i n u o u s ( i . e . p a r a m e t r i c a l l y C

) f o r a l l i = 0 , . . . , n - 1 . I n m a n y p r a c t i c a l a p p l i c a t i o n s the composed map will also be C

continuous and, for simplicity, w e a s s u m e t h i s c a s e h e r e . I n p a r t i c u l a r , C

c o n t i n u i t y o f t h e c o m p o s e d m a p a t t h e c o r n e r ( 0 , 1 ) g i v e s t h e " c o mp a t i b i l i t y c o n d i t i o n s "

. 2 2 j 1 , j 0 ), 1 , 1 1 ( q i j , j j ) 1 ( ) 1 , 0 i ( j q ,

j

= −

∂

− < <

∂ (3. 3)

L e t Ω

⊂ I R

b e a n n - s i d e d p o l y g o n w i t h s i d e s o f l e n g t h u n i t y , c e n t r e t h e origin 0, vertices X

, i=0,...,n-1 and edges E

, i =0,...,n-1, parameterized by

1 . sX i X i ) s 1 ( ) s i (

E = − + + (3.4)

T h e n , f o llowing the approach of [Charrot and Gregory '84], our polygonal p a t c h p : Ω

→ I R

w i l l t a k e t h e f o r m

p , x , 1

n 0 i

) x i ( p ) x i ( w )

x (

p − ∈ Ω

=

= ∑ ^(3.5)

w h e r e p

: Ω

→ I R

i s a p a r a m e t r i c s u r f a c e p a t c h i n t e r p o l a n t w h i c h w i l l b e c o n s t r u c t e d such that it has GC

joins with q

along the edge E

and q

along t h e e d g e E _i .

T h e w e i g h t s w

: I R

→ I R

a r e C

f u n c t i o n s c h o s e n s u c h t h a t p ,

x for , 0 ) x i ( w , 1 1

n 0 i

) x i (

w = ≥ ∈ Ω

−

∑ = (3.6)

a n d

. 1 n ,..., 0 k , 1 i ,i k for , 2 , 1 , 0 j , i 0 k E

j w = = ≠ + = −

∂ (3.7)

T h u s , t o i n v e s t i g a t e t h e G C

j o i n o f p w i t h q

a l o n g t h e e d g e E

, w e m a y w r i t e p i n t h e f o r m

) x i ( r ) x 1 ( p i ) x 1 ( w i ) x i ( p ) x i ( w ) x (

p = + + + + (3.8)

w h e r e

. 2 , 1 , 0 j , i 0 i E

j r = =

∂ (3.9)

-6-

N o w p

a n d p

w i l l e a c h h a v e GC

j o i n s w i t h t h e p a t c h q

a l o n g t h e e d g e E

a n d i f t h e GC

j o i n s a r e c h o s e n t o b e i d e n t i c a l , t h e n , b y P r o p o s i t i o n 3 , t h e c o n v e x c o m b i n a t i o n p a t c h p w i l l h a v e a G C

j o i n w i t h t h e r e c t a n g u l a r p a t c h c o m p l e x . W e a r e t h u s c o n c e r n e d w i t h c o n s t r u c t i n g t h e c o m p o n e n t p a t c h e s

3 p

i

: I R

p Ω → t o s a t i s f y s u c h s p e c i a l G C

c o n d i t i o n s . T h i s c o n s t r u c t i o n i n v o l v e s t h e u s e o f c o o r d i n a t e s y s t e m s ( " c o o r d i n a t e c h a r t s " ) ( s

, t

) , i = 0 , . . . , n - 1 , d e f i n e d in the following section.

4. Coordinate Charts for Interpolation

Let Z _i be the point of intersection of the boundary segments E _{i − 1} and E _{i + 1} . Then

Z i i / Z , i x X : ) x i (

d = < − > (4.1) is the perpendicular distance of x ∈ Ω

from the side E

, where

<x,y> : = x

y

+x

y

(4.2) is t h e E u c l i d e a n s c a l a r p r o d u c t o f x = ( x

, x

) ∈ I R

a n d y = ( y

, y

) ∈ I R

.

C o o r d i n a t e charts are then defined by

.

) x ( d ) x ( d

) x ( , d

) x ( d ) x ( d

) x ( : d

)) x ( t ), x ( s ( : ) x (

i 2

i i 1

i 1

i 1 i i

i

i ⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛

+

= +

= φ

−

− +

− (4.3)

The coordinate chart φ _i corresponds to central projections from Z

and

1

Z

. T h u s E

( s

) i s t h e p o i n t o f i n t e r s e c t i o n o f t h e l i n e f r o m Z

t o x w i t h t h e e d g e E

a n d E

( 1 − t

) is the intersection of the line from Z

to x with

see Figure 1. The interpolants p

: Ω

→ I R

will be constructed as

1

E

, x )), x ( t ), x ( s ( p ) x ( p ) x (

p

=

o φ

=

∈ Ω

(4.4) where p

: [ 0 , 1 ]

→ I R

is chosen to have GC

joins with q

on =0 and t

q

on = 0. GC s

continuity of with p

q

and q

is then guaranteed, since p

is related to through the coordinate chart diffeomorphism p

φ

(see Proposition 1).

T h e e x p l i c i t c o n s t r u c t i o n o f p

( a n d h e n c e p

) i s g i v e n i n t h e n e x t s e c t i o n . H o w e v e r , i n o r d e r to make use of Proposition 3, we first relate the derivative of p

= p

o φ

along the direction E

( s ) − Z

with the appropriate derivative

of in the coordinate chart. p

-7- The chain rule gives

p i E ( s ) ( E i ( s ) Z i ) p i ( s , 0 ) E ( s ) ( E i ( s ) Z i )

i

i

− = ∂ ∂ φ −

∂ (4.5)

where

0 . )) t Z i ) s i ( E ( t ) s i ( E i ( ) t

Z i ) s i ( E ) ( s ( E

i

φ + − =

∂

= ∂

− φ

∂ (4.6)

After some calculation, which for brevity is omitted, we obtain ]) c 2 ) 1 t ( 2 s c 4 /[

t , s ( i )) Z ) s i ( E ( t ) s i ( E

i ( + − = + +

φ (4.7)

where

) n / 2 cos(

c = π . (4.8) Hence

p i ( s , 0 ) ( 0 , 1 ),

) s ( ) 1 Z i ) s i ( E ) ( s ( E p i

i

∂

= γ

−

∂ (4.9) where

. c 2 2 s c 4 ) s

( = +

γ (4.10) Similarly

( (4.11) ).

0 , 1 ) ( s , 0 ( 1 p i ) s 1 ( ) 1 Z ) s ( E ) ( s i ( 1 E

p i

∂ +

−

= γ + −

∂

It should be noted that (4.9) and (4.11) describe the fact that differentiation along the direction (0,1) in the coordinate chart φ _i corresponds, with appro- p r i a t e s c a l i n g , t o d i f f e r e n t i a t i o n a l o n g ( 1 , 0 ) i n t h e c o o r d i n a t e c h a r t φ _{i+ 1} . I t i s t h i s o b s e r v a t i o n t h a t a l l o w s u s t o m a k e u s e o f P r o p o s i t i o n 3 .

5. The Polygonal Patch Interpolant

W e a s s u m e C

p a r a m e t r i c c o n t i n u i t y o f t h e r e c t a n g u l a r p a t c h c o m p l e x , i = 0 , . . . , n - 1 , ( s e e S e c t i o n 3 ) a n d d e f i n e p

: [ 0 , 1 ]

→ I R

b y t h e B o o l e a n q

s u m T a y l o r i n t e r p o l a n t

∑ =

− −

∂ β

= 2 0 j

) 1 , t 1 1 ( q i j , ) 0 t j (

!j s j )

t , s i ( p

0 , j q i ( s , 1 )

2 0 j

) s j (

!j

t j ∂

= β

+ ∑

∑ ∑

= =

∂

− 2 0 j

2 0 j

).

1 , 0 i ( j q ,

! j j 2 1 ! j

t j s j

1 2

2 1 2

1

(5.1)

-8-

Here, β is to be a strictly positive function chosen such that p i = p _i o φ _i

a n d p

= p

o φ

h a v e i d e n t i c a l G C

j o i n s w i t h q _i . W e f i r s t a s s u m e t h a t ,

0 ) 0 .. ( ) 0 . ( , 1 ) 0

( = β = β =

β (5.2)

and then has the interpolation properties: p

, 2 , 1 , 0 j ), 1 , t 1 1 ( q i j , ) 0 t j ( ) t , 0 i ( 0 p

,j = β ∂ − − =

∂ (5.3)

. 2 , 1 , 0 j ), 1 , s i ( j q , ) 0 s j ( ) 0 , s i ( j p ,

0 = β ∂ =

∂ (5.4)

T h e c o n d i t i o n s ( 5 . 3 ) a n d ( 5 . 4 ) m e a n t h a t p

= p

o φ

j o i n s q

a n d q

w i t h GC

c o n t i n u i t y . W e n o w s h o w t h a t β c a n b e c h o s e n s o t h a t t h e G C

j o i n s a r e identical and hence, by Proposition 3 , p will have a GC

join with the rectangular patch complex.

F r o m ( 4 . 9 ) , ( 4 . 1 1 ) a n d t h e i n t e r p o l a t i o n p r o p e r t i e s ( 5 . 3 ) , ( 5 . 4 ) w e O b t a i n

( s ) 0 , 1 q i ( s , 1 ) )

s ) ( Z ) s ( E ) ( s i ( i E

p

∂

γ

= β

∂ −

( 1 s ) 0 , 1 q i ( s , 1 ) )

s 1 ) ( Z ) s ( E ) ( s i ( 1 E

p i

∂

− γ

−

= β + −

∂

T h u s , f o r a n i d e n t i c a l G C

j o i n , w e r e q u i r e t h a t ), s 1 ( / ) s 1 ( ) s ( / ) s

( γ = β − γ −

β (5.5)

w h e r e

. c 2 2 s c 4 ) s

( = +

γ (5.6)

Hence

) s ( / ) s ( ) s

( = γ α

β (5.7)

s a y , w h e r e f r o m ( 5 . 2 ) , ( 5 . 5 ) , a n d ( 5 . 6 ) , α ( s ) m u s t b e c h o s e n s u c h t h a t a n d α ( s ) = α ( 1 − s )

. 0 ) 0 .. ( 2 , c 4 ) 0 . ( , c 2 ) 0

( = α = α =

α

Q u i n t i c H e r m i t e i n t e r p o l a t i o n t h e n p r o v i d e s t h e d e f i n i t i o n

( ) s 4 c 2 ⎜ s 4 2 s 3 s ⎟ ⎠ ⎞ + 2 c ,

⎝ ⎛ − +

=

α (5. 8)

t h i s b e i n g a p o s i t i v e c o n c a v e f u n c t i o n o n 0 ≤ s ≤ 1 . T h i s t o g e t h e r w i t h ( 5 . 7 ) a n d ( 5 . 6 ) d e f i n e s a n a p p r o p r i a t e function β ( s ) .

W e c a n n o w s u m m a r i z e t h e p o l y g o n a l p a t c h i n t e r p o l a n t a s

-9-

)), s i ( t , 1

n 0 i

) x i ( s i ( p ) x i ( w )

x (

p ∑ ⁻

=

= (5.9)

w h e r e p

i s d e f i n e d b y ( 5 . 1 ) a n d t h e c o o r d i n a t e s ( s

( x ), t

( x )) a r e d e f i n e d b y ( 4 . 3 ) . A n a p p r o p r i a t e d e f i n i t i o n o f t h e w e i g h t i n g f u n c t i o n s i s

⎥ ⎥ . (5.10)

⎦

⎤

⎢ ⎢

⎣

⎡ −

= ≠ + +

≠

= ∏ ⁿ ∑ ∏ ¹

0

k j k , k 1 3 j d /

1 i, i j

3 j d )

x i ( w

6. Shape Control

E q u a t i o n s ( 5 . 7 ) a n d ( 5 . 8 ) p r o v i d e o n e o f i n f i n i t e l y m a n y p o s s i b l e d e f i n i t i o n s f o r t h e s c a l i n g f u n c t i o n β ( s ) = γ ( s ) / α ( s ). Defining

3 , ) s 1 3 ( i s ) s i ( where ),

s i ( / ) s ( ) s

i ( = γ α α = ν −

β (6.1)

p r o v i d e s a m e t h o d o f s h a p e c o n t r o l a c r o s s t h e b o u n d a r i e s E

, f o r g i v e n r e a l p a r a m e t e r s ν _i . T h e t e r m s β ( t ) a n d β ( s ) i n ( 5 . 1 ) m u s t t h e n b e r e p l a c e d b y β

( t ) and β

( s ) respectively.

A n o t h e r p o s s i b i l i t y f o r s h a p e c o n t r o l s u g g e s t e d i n [ G r e g o r y a n d H a h n ‘ 8 7 ] is to add a term of the form

∏ ⁻

= 1 n

0 i

3 i d ) x (

r (6.2) t o t h e p o l y g o n a l p a t c h i n t e r p o l a n t p . T h i s t e r m w i l l n o t a f f e c t t h e GC

c o n t i n u i t y a l o n g t h e b o u n d a r y a n d r ( x ) m a y t h u s b e u s e d t o c o n t r o l t h e s h a p e

o f t h e p o l y g o n a l p a t c h i n t e r i o r .

-10- References

C h a r r o t , P . a n d G r e g o r y , J . A . ( 1 9 8 4 ) , A p e n t a g o n a l s u r f a c e p a t c h f o r c o m p u t e r a i d e d g e o m e t r i c d e s i g n , C o m p u t e r A i d e d G e o m e t r i c D e s i g n 1 , 8 7 - 9 4 .

Gregory, J.A. and Hahn, J.M. (1986), Geometric continuity and convex c o m b i n a t i o n p a t c h e s , t o a p p e a r i n C o m p u t e r A i d e d G e o m e t r i c D e s i g n .

Gregory, J.A. and Hahn, J.M. (1987), Polygonal patches of high order c o n t i n u i t y , i n t e r n a l r e p o r t T R / 0 1 / 8 7 , D e p t . M a t h s . & S t a t . , B r u n e l U n i v e r s i t y .

Hahn, J.M. (1987), Geometric continuous patch complexes, these proceedings, s e e

a l s o i n t e r n a l r e p o r t T R / 0 5 / 8 7 , D e p t . M a t h s . & S t a t . , B r u n e l U n i v e r s i t y .